The aim of this article is to study (additively) indecomposable algebraic integers  of biquadratic number fields and universal totally positive quadratic forms with coefficients in  . There are given sufficient conditions for an indecomposable element of a quadratic subfield to remain indecomposable in the biquadratic number field . Furthermore, estimates are proven which enable algorithmization of the method of escalation over . These are used to prove, over two particular biquadratic number fields ℚ ( √ 2,, a lower bound on the number of variables of a universal quadratic forms. K E Y W O R D S biquadratic number field, indecomposable integer, universal quadratic form M S C ( 2 0 1 0 ) 11E12, 11E25, 11R04 540

We study totally positive definite quadratic forms over the ring of integers $\mathcal {O}_K$ of a totally real biquadratic field $K=\mathbb {Q}(\sqrt {m}, \sqrt {s})$ . We restrict our attention to classic forms (i.e. those with all non-diagonal coefficients in $2\mathcal {O}_K$ ) and prove that no such forms in three variables are universal (i.e. represent all totally positive elements of $\mathcal {O}_K$ ). Moreover, we show the same result for totally real number fields containing at least one non-square totally positive unit and satisfying some other mild conditions. These results provide further evidence towards Kitaoka's conjecture that there are only finitely many number fields over which such forms exist. One of our main tools are additively indecomposable elements of $\mathcal {O}_K$ ; we prove several new results about their properties.

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In this article, the standard correspondence between the ideal class group of a quadratic number field and the equivalence classes of binary quadratic forms of given discriminant is generalized to any base number field of narrow class number one. The article contains an explicit description of the correspondence. In the case of totally negative discriminants, equivalent conditions are given for a binary quadratic form to be totally positive definite.